We revisit three fundamental problems in algorithms under uncertainty: the Secretary Problem, Prophet Inequality, and Stochastic Probing, each subject to general downward-closed constraints. When elements have binary values, all three problems admit a tight $\tildeΘ(\log n)$-factor approximation guarantee. For general (non-binary) values, however, the best known algorithms lose an additional $\log n$ factor when discretizing to binary values, leaving a quadratic gap of $\tildeΘ(\log n)$ vs. $\tildeΘ(\log^2 n)$. We resolve this quadratic gap for all three problems, showing $\tildeΩ(\log^2 n)$-hardness for two of them and an $O(\log n)$-approximation algorithm for the third. While the technical details differ across settings, and between algorithmic and hardness proofs, all our results stem from a single core observation, which we call the Big-Decisions-First Principle: Under uncertainty, it is better to resolve high-stakes (large-value) decisions early.

翻译：我们重新审视不确定性算法中的三个基本问题：Secretary 问题、Prophet 不等式和随机探测问题，每个问题都受到一般向下封闭约束的限制。当元素具有二元值时，这三个问题都承认一个紧的 $\tildeΘ(\log n)$ 因子近似保证。然而，对于一般（非二元）值，已知最优算法在离散化为二元值时额外损失 $\log n$ 因子，导致在 $\tildeΘ(\log n)$ 和 $\tildeΘ(\log^2 n)$ 之间存在二次间隙。我们解决了这三个问题的二次间隙，证明了其中两个问题的 $\tildeΩ(\log^2 n)$-难度，并为第三个问题给出了 $O(\log n)$-近似算法。尽管技术细节在不同设置、算法和难度证明之间有所不同，但我们的所有结果都源于一个核心观察，我们称之为“大决策优先”原则：在不确定性下，优先解决高风险（大值）的决策更为有利。